Theorem sibfima 34531

Description: Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)

Hypotheses

Ref	Expression
sitgval.b	⊢ 𝐵 = (Base‘𝑊)
sitgval.j	⊢ 𝐽 = (TopOpen‘𝑊)
sitgval.s	⊢ 𝑆 = (sigaGen‘𝐽)
sitgval.0	⊢ 0 = (0g‘𝑊)
sitgval.x	⊢ · = ( ·𝑠 ‘𝑊)
sitgval.h	⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1	⊢ (𝜑 → 𝑊 ∈ 𝑉)
sitgval.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
sibfmbl.1	⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))

Assertion

Ref	Expression
sibfima	⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))

Proof of Theorem sibfima

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sibfmbl.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
2		sitgval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
3		sitgval.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊)
4		sitgval.s	. . . . 5 ⊢ 𝑆 = (sigaGen‘𝐽)
5		sitgval.0	. . . . 5 ⊢ 0 = (0g‘𝑊)
6		sitgval.x	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
7		sitgval.h	. . . . 5 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
8		sitgval.1	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
9		sitgval.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
10	2, 3, 4, 5, 6, 7, 8, 9	issibf 34526	. . . 4 ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
11	1, 10	mpbid 233	. . 3 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
12	11	simp3d 1150	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
13		sneq 4566	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
14	13	imaeq2d 6013	. . . . 5 ⊢ (𝑥 = 𝐴 → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝐴}))
15	14	fveq2d 6832	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑀‘(◡𝐹 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝐴})))
16	15	eleq1d 2824	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)))
17	16	rspcv 3556	. 2 ⊢ (𝐴 ∈ (ran 𝐹 ∖ { 0 }) → (∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)))
18	12, 17	mpan9 511	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ∖ cdif 3880  {csn 4556  ∪ cuni 4839  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   “ cima 5622  ‘cfv 6486  (class class class)co 7357  Fincfn 8884  0cc0 11030  +∞cpnf 11168  [,)cico 13292  Basecbs 17171  Scalarcsca 17215   ·_𝑠 cvsca 17216  TopOpenctopn 17376  0_gc0g 17394  ℝHomcrrh 34186  sigaGencsigagen 34331  measurescmeas 34388  MblFnMcmbfm 34442  sitgcsitg 34522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-sitg 34523

This theorem is referenced by:  sibfinima  34532  sitgfval  34534  sitgclg  34535